Topological bifurcations and reconstruction of travelling waves

Abstract

This paper is devoted to periodic traveling waves solving Lie–Poisson equations based on the Virasoro group. We show that the reconstruction of any such solution can be carried out exactly, regardless of the underlying Hamiltonian (which need not be quadratic), provided the wave belongs to the coadjoint orbit of a uniform profile. Equivalently, the corresponding “fluid particle motion” is integrable. Applying this result to the Camassa–Holm equation, we express the drift of particles in terms of parameters labeling periodic peakons and exhibit orbital bifurcations: points in parameter space where the drift velocity varies discontinuously, reflecting a sudden change in the topology of Virasoro orbits.